Bracket Cost

2000ms

262144K

Description:

Daemon Targaryen decided to stop looking like a Metin2 character. He turned himself into the most beautiful thing, a bracket sequence.

For a bracket sequence, we can do two kind of operations:

Select one of its substrings$$$^\dagger$$$ and cyclic shift it to the right. For example, after a cyclic shift to the right, "(())" will become ")(()";
Insert any bracket, opening '(' or closing ')', wherever you want in the sequence.

We define the cost of a bracket sequence as the minimum number of such operations to make it balanced$$$^\ddagger$$$.

Given a bracket sequence $$$s$$$ of length $$$n$$$, find the sum of costs across all its $$$\frac{n(n+1)}{2}$$$ non-empty substrings. Note that for each substring we calculate the cost independently.

$$$^\dagger$$$ A string $$$a$$$ is a substring of a string $$$b$$$ if $$$a$$$ can be obtained from $$$b$$$ by deletion of several (possibly, zero or all) characters from the beginning and several (possibly, zero or all) characters from the end.

$$$^\ddagger$$$ A sequence of brackets is called balanced if one can turn it into a valid math expression by adding characters $$$+$$$ and $$$1$$$. For example, sequences "(())()", "()", and "(()(()))" are balanced, while ")(", "(()", and "(()))(" are not.

Input:

Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^5$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the length of the bracket sequence.

The second line of each test case contains a string $$$s$$$, consisting only of characters '(' and ')', of length $$$n$$$ — the bracket sequence.

It is guaranteed that sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output:

For each test case, print a single integer — the sum of costs of all substrings of $$$s$$$.

Sample Input:

5
1
)
4
)()(
3
())
5
(((((
10
)(())))())

Sample Output:

Note:

In the first test case, there is the only substring ")". Its cost is $$$1$$$ because we can insert '(' to the beginning of this substring and get a string "()", that is a balanced string.

In the second test case, the cost of each substring of length one is $$$1$$$. The cost of a substring ")(" is $$$1$$$ because we can cyclically shift it to right and get a string "()". The cost of strings ")()" and "()(" is $$$1$$$ because its enough to insert one bracket to each of them. The cost of substring ")()(" is $$$1$$$ because we can cyclically shift it to right and get a string "()()". So there are $$$4 + 2 + 2 + 1 = 9$$$ substring of cost $$$1$$$ and $$$1$$$ substring of cost $$$0$$$. So the sum of the costs is $$$9$$$.

In the third test case,

"(", the cost is $$$1$$$;
"()", the cost is $$$0$$$;
"())", the cost is $$$1$$$;
")", the cost is $$$1$$$;
"))", the cost is $$$2$$$;
")", the cost is $$$1$$$.

So the sum of the costs is $$$6$$$.

Informação

Provedor Codeforces

Código CF1750E